Neighborhoods of adjacent cells are overlapping for k−1=2{\displaystyle k-1=2} cells. There are |S|k−1=4{\displaystyle |S|^{k-1}=4} different overlaps 00,01,10,11{\displaystyle {00,01,10,11}} or written in compact form 0,1,2,3{\displaystyle {0,1,2,3}}.

The quiescent background is an infinite sequence with period α=0¯{\displaystyle \alpha ={\overline {0}}} of length |α|=1{\displaystyle |\alpha |=1} cell.

There are always exactly two preimages for this background independently on the length of the sequence from a single cell to infinity (see the preimage network). The left and right boundary vectors are equal.

The ether background is an infinite sequence with period α=00010011011111¯{\displaystyle \alpha ={\overline {00010011011111}}} of length |α|=14{\displaystyle |\alpha |=14} cells. This is the prevailing background emerging from a random initial configuration.

The number of preimages of the ether configuration increases exponentially with the sequence length l{\displaystyle l} going to infinity. A circular lattice is used to calculate the number of preimages of the period α{\displaystyle \alpha }.

Because of the exponential growth the boundary vector does not represent the number of preimages of the whole infinite background but only weights derived from the period's preimages. The value of the boundary vector depends on the position inside the period, in the next table vectors are columns for each of the 14 positions.